GFR Wrote:(e) In the first hypothesis (continuous line), the curves of the real values passing through all the odd or the even points defined by y = b # n are just "envelopes" of the actual almost sinusoidal real line oscillating around a mid-value. The upper and lower envelopes may very well be continuous, but the almost periodicity of the "evelopped lines" describing y = b # x will be always 2 (odd/even alternations). No discontinuous jumps between max and min y are detectable. "Tetratio non facit saltus".

(e) In the second hypothesis (dusty distribution), we have not only sudden jumps between any dx variations but also fuzzy point distributions, which would suggest us to give up and do other things.

I also vote for now clearly defined e) but with additional imaginary dimension or "hyperdimensions " where it all happens. Cylinder. Any 2 points on real axis are connected by a line (curve) which leaves real axis and goes to infinity and returns on next point, infinitesimally close to previous. So a point on real axis is just such an infinite but connected line looked at from end. And, there are 2 possible rotation directions of these lines. And, what about third, x^(1/x) which tends to be continuous without leaving real xy plane? it seems like that one is rotating along its own axis, experiencing torsion.

I had a question which I did could not find a satisfactory answer-if n= 3 , x^n has 1 value, x^1/n = 3 values, 2 of them complex.

How many values has x^(1/x) when x= rational ( lime m/n m is not 1) irrational, transcendental, complex, i?

For me it is obvious that mathematics have to be defined in terms of i from which we can obtain via "backtetration" e^(pi/2) - not in terms of counting and integer numbers. From the other end, so to say, or rather, from both other ends simultaneously.

Ivars Wrote:Any 2 points on real axis are connected by a line (curve) which leaves real axis and goes to infinity and returns on next point, infinitesimally close to previous.

Ivars

What? This has nothing to do with what I am trying to say. Are you joking?

GFR

No... It is my intuitive understanding of continuity. You were talking about sinusoids in real plane connecting h even h odd , or perhaps I misunderstood?

I just took the sinusoids out of real xy plane. Based on the fact that tetration tends to take real numbers away from purely real , why can not h(x) < e^-e when extented to real tetration parameter (so that between n odd and n interger as n-> infinity we have n real such that as z-> infinity but is not integer, x^^z = imaginary or complex?

Ivars Wrote:Any 2 points on real axis are connected by a line (curve) which leaves real axis and goes to infinity and returns on next point, infinitesimally close to previous.

What? This has nothing to do with what I am trying to say. Are you joking?

GFR

No... It is my intuitive understanding of continuity. You were talking about sinusoids in real plane connecting h even h odd , or perhaps I misunderstood?

I meant the yx plane where we might "see" the sections of y = b # x = b-tetra-x, for b = const. and of their "envelopes". Real numbers h (hsup, hinf) are the limits, for x -> +oo of these envelopes. These "sections" should show, in my opinion, oscillatory behaviours between the ysup and yinf values detectable for any x = n (integer) even and odd, respectively. They must also show, always according to my point of view, continuous line connections between these up/down points. The pseudo-period should be 2 (I mean: 1 + 1).

Ivars Wrote:I just took the sinusoids out of real xy plane. Based on the fact that tetration tends to take real numbers away from purely real , why can not h(x) < e^-e when extented to real tetration parameter (so that between n odd and n interger as n-> infinity we have n real such that as z-> infinity but is not integer, x^^z = imaginary or complex?

We are not on the same track. What I am just trying to say is that the above-mentioned hypothetical oscillatory connections between the up/down points should be continuous and decreasing and could be represented as the projections of y* (complex y) against x, on the real yx plane. Complex y (y*) would then be representable, qualitatively, by a vector y* = |y*|. e^i.Pi.x , rotating according to an imaginary i.Pi.x angular dimension, giving:
y* = |y*|(cos Pi.x + i sin Pi.x)
Supposing |y*| constant, and I am sure that it is not, the yx real projection would then be:
Re[y*] = |y*|. cos Pi.x , with period x = 2.
Nothing goes to infinity and comes back after an infinitesimal length. However, for b < e ^(-e), when x -> +oo the gap netween ysup and yinf does not vanish and become hsup - hinf, producing an indetermination between the two extremes. However, in this model, also the asymptotic value of y*, for x -> +oo, should remain complex. The "yellow zone" or "transition area" would then be the graph of this asymptotic real gap, against b, plotted on the yb plane.

By the way, perhaps you also meant: b < e ^(-e).

But, these is only the product of my imagination. Things could be different. Who knows. But, perhaps, we are annoying the other Participants.

I vote for (g), moving this thread to "General Discussion" since it is not very mathematical.

Ivars, if you are going to be talking about anything, you have to use words and symbols we understand, otherwise your insights will be without meaning, and without content. If you want to talk about imaginary infinitesimals, use , and if you want to talk about lines "going to infinity and back", then you should research the proper terminology for discussing these concepts, and refer to a geodesic of the Riemann sphere instead. As your comments stand, have very little meaning, and only vauge definitions behind them. Please try and be more precise so that we can more clearly understand you, otherwise, anything you say could be true or false, and there wouldn't be any way to know.

I am afraid I have already made too much noise so when I will really have something to show, no one will even notice. I take a small break I am investigating superroot right now which actually seems to be very simple but very rich operation. Barrow said (not exact quote):

if x^x=1/2 has no solutions in real and complex numbers, it may lead to creation of new numbers just to fill that gap. Has anyone already tried to define such numbers?